Answer key: Area & Perimeter Worksheet (With Composite Shapes)

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Goal: Find the area and perimeter of common shapes, explain how area and perimeter differ, and find the area of a composite figure by adding and subtracting parts.

Connects to Common Core 6.G.A.1 (Find the area of triangles, special quadrilaterals and polygons by composing them into rectangles or decomposing them into triangles and other shapes.); 7.G.B.4 (Know the formulas for the area and circumference of a circle and use them to solve problems.); 7.G.B.6 (Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects.)

Completed results table

Values are the readouts of the tool (Area calculator) for each setting.

Answers

1. The two rectangles have the same perimeter. Do they have the same area? What does this tell you?

   No. Both perimeters are 26 cm, but the 8 × 5 rectangle has an area of 40 cm² and the 12 × 1 rectangle only 12 cm². Perimeter (the distance around) and area (the space inside) measure different things. For the same perimeter, the closer a rectangle is to a square, the more area it encloses.

2. Compare the triangle (base 8, height 5) with the 8 × 5 rectangle. Explain the relationship.

   The triangle’s area is 20 cm², exactly half of the rectangle’s 40 cm². A triangle is half of a rectangle with the same base and height: A = ½bh.

3. Find the trapezoid’s area by hand with A = ½(a + b)h.

   ½ × (8 + 5) × 4 = ½ × 13 × 4 = 26 cm². It has the same area as a rectangle 6.5 cm wide (the average of the parallel sides) and 4 cm tall.

4. For the circle with radius 5, find the area and circumference by hand (use π ≈ 3.14159). What happens to each if the radius doubles to 10?

   A = πr² = 25π ≈ 78.54 cm² and C = 2πr = 10π ≈ 31.416 cm. With a radius of 10, the circumference doubles to about 62.832 cm, but the area becomes 4 times as large: about 314.159 cm².

5. Explain how the composite calculator found the area of the L-shape.

   It adds the 10 × 6 rectangle (60 square units) and subtracts the missing 4 × 3 corner (12 square units): 60 − 12 = 48 square units.

6. Show the working for the rectangle with a semicircle on one end.

   The rectangle is 10 × 6 = 60, and the semicircle is ½ × π × 3² ≈ 14.137. Total ≈ 60 + 14.137 = 74.137 square units.